Schedule for: 23w5134 - Arithmetic Aspects of Deformation Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Potentially crystalline Emerton-Gee stacks are finite type p-adic formal stacks interpolating Kisin's potentially crystalline Galois deformation rings, and are expected to play a critical role in the emerging categorical p-adic Langlands program. However, even in the simplest non-trivial case of rank 2 tame potentially Barsotti-Tate stacks, these stacks exhibit extremely complicated behaviors when the tame type is non-generic, making them hard to understand. I will describe a general approach to construct algebraic models for these stacks for tame inertial type and p-small Hodge-Tate weights when p is large. Time permitting, I will talk about several applications, such as the geometric Breuil-Mezard conjecture for generic tame types with close to optimal genericity (joint work in progress with T. Feng), and a conceptualization of computations of Caruso-David-Mezard in the rank 2 tame potentially Barsotti-Tate case (joint work with A. Mezard and S. Morra).

A categorical perspective, which has long been the norm in the geometric Langlands program, has recently emerged in the arithmetic Langlands program as well. I will discuss some of the ideas related to this perspective, including the Fargues--Scholze conjecture; conjectures and results (due variously to Ben-Zvi--Chen--Helm--Nadler, Hellmann, Zhu, as well as the speaker, Andrea Dotto, and Toby Gee) regarding fully faithful functors from categories of representations of p-adic reductive groups to categories of coherent sheaves on stacks of Langlands parameters; and conjectural descriptions of the cohomology of locally symmetric congruence quotients (e.g. Shimura varieties) in terms of these categorical ideas. I will try to emphasize the example of modular curves, and the connection to more classical perspectives on modularity, Fontaine--Mazur, and related topics.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Suppose that $F$ is a totally real field and $v$ a place dividing $p$. A (nice) 2-dimensional mod $p$ representation $\rho : \mathrm{Gal}(\overline F/F) \to \mathrm{GL}_2(\overline{\mathbb F}_p)$ cuts out a mod $p$ smooth representation $\pi$ of $\mathrm{GL}_2(F_v)$ in the cohomology of a Shimura curve. In previous work we constructed a multivariable $(\varphi,\Gamma)$-module $D_A(\pi)$, where $\Gamma = \mathcal O_K^\times$. We show that $D_A(\pi) \cong D_A^\otimes(\rho|_{G_{F_v}})$, where $G_{F_v}$ is a decomposition group at $v$ and $D_A^\otimes$ is a new functor constructed using perfectoid techniques. This is joint work with C. Breuil, Y. Hu, S. Morra, and B. Schraen.

In his seminal work on modular curves and the Eisenstein ideal, Mazur studied the existence of congruences between certain Eisenstein series and newforms, proving that Eisenstein ideals associated to weight 2 cusp forms of prime level are locally principal. In this talk, we'll explore generalizations of Mazur's result to squarefree level, focusing on recent work, joint with P. Wake and C. Wang-Erickson, about a non-optimal level N that is the product of two distinct primes and where the Galois deformation ring is not expected to be Gorenstein. First, we will outline a Galois-theoretic criterion for the deformation ring to be as small as possible, and when this criterion is satisfied, deduce an R=T theorem. Then we'll discuss some of the techniques required to computationally verify the criterion.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Fix a weight 2 CM modular form with trivial nebentypus and level $\ell^2$ for some prime $\ell$, say $f$. How many forms in $S_2(\Gamma_0(\ell^2))$ are congruent to $f$ modulo a prime $p$? We will sketch a proof that when $\ell \equiv -1 \bmod p$, this number is always divisible by $p$. Such an $f$ is an example of a modular form that we call “vexing at $\ell$ modulo $p$”, and the $p$-divisibility phenomenon is true for all such vexing forms (at least if $p \geq 5$). This is work in progress with Robert Pollack and Preston Wake.

Let $p$ be a prime and $F$ a $p$-adic local field with absolute Galois group $G_F$. Let $\CO$ be the ring of integers of a $p$-adic field $L\supset F$ with residue field $\BF$, and consider a continuous representation $\bar\rho\colon G_F\to\GL_d(\BF)$. Denote by $R_{\bar\rho}^\square$ the universal ring for liftings $\rho_R\colon G_F\to \GL_d(R)$ of $\bar\rho$ to complete noetherian local $\CO$-algebras $R$ with residue field~$\BF$. Let $\mu_{p^\infty}(F)$ be the set of $p$-power roots of unity of~$F$. I will report on joint work with Ashwin Iyengar and Vytautas Pa\v{s}k\=unas which establishes foundational properties of $R_{\bar\rho}^\square$ for any $\bar\rho$. It is a complete intersection ring and flat over $\CO$ of relative dimension $d^2[F:\BQ_p]+d^2$, the irreducible components of $\Spec R_{\bar\rho}^\square$ are in bijection with $\mu_{p^\infty}(F)$ and each of them contains a dense set of crystalline characteristic zero points.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will report on an ongoing joint work with Julian Quast on deformation rings of Galois representations of p-adic fields valued in reductive groups.

Let $(G, X)$ be a Shimura datum, and let $K$ be a compact open subgroup of $G(\mathbb{A}_f)$. One hopes that under mild assumptions on $G$ and $K$, the points of the Shimura variety $Sh_K(G, X)$ form a family of motives; in abelian type this is well-understood, but in non-abelian type it is completely mysterious. I will discuss joint work with Christian Klevdal showing that for exceptional Shimura varieties the points (over number fields, say) at least yield compatible systems of l-adic representations (that should be the l-adic realizations of the conjectural motives).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

It is widely understood and pervasively applied that an appropriate 1st cohomology group expresses first order deformations, while obstructions arise from a 2nd cohomology group. But what data controls these obstructions? In this talk, we will explain what an A-infinity algebra is, where it comes from, and how an A-infinity structure naturally arises as a convenient package to use to produce answers to this question in the context of equal-characteristic deformations. Then we will describe some applications to Galois deformation theory and the modulo p representation theory of p-adic groups.

Given a $Z_p$ extension $F_\infty$ of a number field $F$, we define and study (under Calegari-Geraghty type assumptions) classical and derived deformation rings over $F_\infty$ of an ordinary automorphic residual Galois representation. We proceed axiomatically by working with an arbitrary connected reductive group, assuming that the Galois representation associated to a cuspidal automorphic representation is defined and satisfies the conjectured local-global compatibilities. This is a joint work with E. Urban.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Tame triple product periods are images of a product of two weight 1 modular forms in the generalised eigenspace attached to a mod p eigenform of weight 2. They can be viewed as a mod p analogue of complex triple product periods, which are related to L-values by the work of Harris-Kudla. In this talk, I will present a conjecture relating triple product periods to global points of elliptic curves. It can be viewed as an elliptic counterpart of a conjecture of Harris-Venkatesh on derived hecke operators. I will sketch a proof of this conjecture for weight 1 dihedral CM forms, relying on the structure of the supersingular module as a module over the hecke algebra of weight 2 modular forms. This is joint work with Henri Darmon.

Given an elliptic curve E, Kolyvagin used CM points on modular curves to construct a system of classes valued in the Galois cohomology of the torsion points of E. Under the conjecture that not all of these classes vanish, he deduced remarkable consequences for the Selmer rank of E. For example, his results, combined with work of Gross-Zagier, implied that a curve with analytic rank one also has algebraic rank one; a partial converse follows from his conjecture. In this talk, I will describe a proof of new cases of Kolyvagin's conjecture. The source of the improvement over previous results is to use level-raising congruences modulo large powers of p (rather than just modulo p); a deformation theoretic input is needed to lift modulo p^n modular eigenforms to characteristic zero. I will explain why this input is necessary and how it is achieved using modularity lifting theorems and techniques of Fakhruddin-Khare-Patrikis.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will outline the construction of nontrivial elements in the Bloch--Kato Selmer group of the symmetric cube of a modular form f, provided that the symmetric cube functional equation of f has sign -1, conditional on Arthur's conjectures for the exceptional group G_2. The construction passes through the p-adic deformation of Eisenstein series for G_2, and I will focus on this aspect of the construction.

Wiles' modularity lifting theorem was the central argument in his proof of modularity of (semistable) elliptic curves over Q, and hence of Fermat's Last Theorem. His proof relied on two key components: his "patching" argument (developed in collaboration with Taylor) and his numerical isomorphism criterion. In the time since Wiles' proof, the patching argument has been generalized extensively to prove a wide variety of modularity lifting results. In particular Calegari and Geraghty have generalized it to the "positive defect case" which includes weight one modular forms and automorphic forms on PLG2 over arbitrary number fields (contingent on some standard conjectures in some cases). The numerical criterion on the other hand has proved far more difficult to generalize, although in situations where it can be used it can prove stronger results than what can be proven purely via patching. In this talk I will present joint work with Srikanth Iyengar and Chandrashekhar Khare which gives a generalization of the numerical criterion to positive defect case (contingent on the same conjectures). This allows us to prove some integral "R=T" theorems at non-minimal levels which are inaccessible by Calegari and Geraghty's method. Our method combines the numerical criterion with patching and generalizes the theory of congruence modules to higher dimensional rings.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

No formal activities planned for Friday

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.